Luogu4717 【模板】快速沃尔什变换 (FWT)

https://www.luogu.com.cn/problem/P4717

参考blog

参考blog2

\(FWT\)

参考\(blog\)非常详细，可以仔细看看

\(or:\)

\[FWT(a)=merge(FWT(a_0),FWT(a_0)+FWT(a_1))\\ a=merge(FWT(a_0),FWT(a_1)-FWT(a_0)) \]

\(and:\)

\[FWT(a)=merge(FWT(a_0)+FWT(a_1),FWT(a_1))\\ a=merge(FWT(a_0)-FWT(a_1),FWT(a_1)) \]

\(xor:\)

\[FWT(a)=merge(FWT(a_0)+FWT(a_1),FWT(a_0)-FWT(a_1))\\ a=merge(\frac{FWT(a_0)+FWT(a_1)}{2},\frac{FWT(a_0)-FWT(a_1)}{2}) \]

\(FWT\)过程中一定要取模！

\(C++ Code:\)

#include<cstdio>
#include<iostream>
#include<algorithm>
#define p 998244353
#define ll long long
#define N (1 << 18)
using namespace std;
int n;
ll inv2,A[N],B[N],a[N],b[N];
ll ksm(ll x,ll y)
{
    ll ans=1;
    while (y)
    {
        if (y & 1)
            ans=ans*x%p;
        x=x*x%p;
        y >>=1;
    }
    return ans;
}
void FWT_Or(ll *a,ll t)
{
    for (int mid=1;mid<n;mid <<=1)
        for (int j=0;j<n;j+=(mid << 1))
            for (int k=0;k<mid;k++)
            {
                ll x=a[j+k],y=a[j+k+mid];
                a[j+k]=x;
                a[j+k+mid]=t*(x+t*y)%p;
            }
}
void FWT_And(ll *a,ll t)
{
    for (int mid=1;mid<n;mid <<=1)
        for (int j=0;j<n;j+=(mid << 1))
            for (int k=0;k<mid;k++)
            {
                ll x=a[j+k],y=a[j+k+mid];
                a[j+k]=(x+t*y)%p;
                a[j+k+mid]=y;
            }
}
void FWT_Xor(ll *a,ll t)
{
    for (int mid=1;mid<n;mid <<=1)
        for (int j=0;j<n;j+=(mid << 1))
            for (int k=0;k<mid;k++)
            {
                ll x=a[j+k],y=a[j+k+mid];
                a[j+k]=(x+y)*((t==1)?1:inv2)%p;
                a[j+k+mid]=(x-y)*((t==1)?1:inv2)%p;
            }
}
int main()
{
    inv2=ksm(2,p-2);
    scanf("%d",&n);
    n=(1 << n);
    for (int i=0;i<n;i++)
        scanf("%lld",&A[i]);
    for (int i=0;i<n;i++)
        scanf("%lld",&B[i]);
    for (int i=0;i<n;i++)
        a[i]=A[i],b[i]=B[i];
    FWT_Or(a,1);
    FWT_Or(b,1);
    for (int i=0;i<n;i++)
        a[i]=a[i]*b[i]%p;
    FWT_Or(a,-1);
    for (int i=0;i<n;i++)
    {
        a[i]=(a[i]%p+p)%p;
        printf("%lld ",a[i]);
    }
    putchar('\n');
    for (int i=0;i<n;i++)
        a[i]=A[i],b[i]=B[i];
    FWT_And(a,1);
    FWT_And(b,1);
    for (int i=0;i<n;i++)
        a[i]=a[i]*b[i]%p;
    FWT_And(a,-1);
    for (int i=0;i<n;i++)
    {
        a[i]=(a[i]%p+p)%p;
        printf("%lld ",a[i]);
    }
    putchar('\n');
    for (int i=0;i<n;i++)
        a[i]=A[i],b[i]=B[i];
    FWT_Xor(a,1);
    FWT_Xor(b,1);
    for (int i=0;i<n;i++)
        a[i]=a[i]*b[i]%p;
    FWT_Xor(a,-1);
    for (int i=0;i<n;i++)
    {
        a[i]=(a[i]%p+p)%p;
        printf("%lld ",a[i]);
    }
    putchar('\n');
    return 0;
}